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1 City Research Online City, University of London Institutional Repository Citation: Linckelmann, M. (1999). Transfer in Hochschild Cohomology of locks of Finite Groups. lgebras and Representation Theory, 2(2), pp doi: /: This is the unspecified version of the paper. This version of the publication may differ from the final published version. Permanent repository link: Link to published version: Copyright and reuse: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. City Research Online: publications@city.ac.uk

2 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS OF FINITE GROUPS Markus Linckelmann July Typeset by MS-TEX

3 2 MRKUS LINCKELMNN 1 Introduction and Notation 1.1 The cohomology of a finite group G with coefficients in a complete discrete valuation ring O having a residue field k of prime characteristic p is the ring H (G,O) = Ext OG (O,O), whereo isconsideredastrivialog module; thatis, with x G acting as identity on O. It is apparent from this definition, that H (G,O) is an invariant of the principal block b 0 of OG (this is the unique primitive idempotent of the center Z(OG) of the group algebra OG of G over O acting as the identity on the trivial OG module). Since in general ap block of Gneed not have an augmentation, this definitiondoes not generalize to arbitrary blocks. There is, however, a p local characterization of H (G,O) as the subring of the cohomology ring H (P,O), where P is a Sylowp subgroup of G, consisting of all elements in H (P,O) whose restriction to Q is stable under the action of N G (Q) on H (Q,O) for every subgroup Q of P (cf. [9, Ch. XII, 10.1]). This characterization now does generalize to an arbitrary p block b of G (this is a primitive idempotent of Z(OG)) by taking for P a defect group of b and replacing N G (Q) by N G (Q,e Q ), where (Q,e Q ) is a b rauer pair contained in (P,e P ) for some fixed choice of a suitable block e P of kc G (P) (cf. [1]). We are going to call the ring thus obtained the cohomology ring of the block b (see section 5 for a precise definition). It is well-known that the usual cohomology ring H (G,O) embeds into the Hochschild cohomology ring HH (OG) through diagonal induction (see 4.5 below). We develop here the machinery to show that the cohomology ring of the block b embeds into the cohomology ring HH (OGb) of the block algebra OGb of b. In section 2 we define for any two symmetric algebras, over a commutative ring R and any bounded complex X of finitely generated left and right projective bimodules a transfer map t X : HH () HH () and study its basic properties. We show in particular, analogously to. Keller s results in [12] on transfer in cyclic homology, that t X depends only on the image of X in the appropriate Grothendieck group. In order to study the multiplicative structure of HH () we define in section 3 the notion of X stable elements in HH (), which are then shown to form a graded subalgebra HH X () of HH (); moreover, we show that under some non degeneracy hypothesis on X there is a normalized transfer T X inducing a surjective R algebra homomorphism HH X () HH X (), where X is the R dual of X. Section 4 is devoted to showing that the transfer maps between HH (RG) and HH (RH), where H is a subgroup of the finite group G, obtained from induction and restriction, are compatible with the usual transfer and restriction maps between the cohomology rings H (G,R) and H (H,R) through the diagonal embeddings of the cohomology rings into the Hochschild cohomology rings of G and H, respectively. In section 5 we give the definition of the cohomology ring of a p block b of a finite group G over the ring O and apply the results of the previous sections in order to show that indeed this cohomology ring embeds into HH (OGb) and that the normalized transfer of a suitable p permutation complex always induces the identity on the image of this embedding; this should be thought of as a generalization of the well-known fact, that if H controls p fusion in G then the restriction from G to H

4 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 3 induces an isomorphism H (G,O) = H (H,O) by the p local characterization of the cohomology rings. y Mislin s theorem [15], the converse of this last statement is true, too, and in fact, a major guideline for developing this material is that we expect there to be some generalization of Mislin s theorem to arbitrary p blocks of finite groups; see remark 5.8 below. 1.2 Notation. ll algebras and rings are associative with unit element, all modules are supposed to be finitely generated unitary, and, if not stated otherwise, left modules If,, C are algebras over a commutative ring R, by an bimodule M we mean a bimodule whose left and right R module structure coincide; that is, we may consider M as 0 module with a b acting on m M as amb, where R a, b, and 0 is the algebra obtained from endowing with the opposite product b.b = b b, b,b. The R dual M = Hom R (M,R) becomes then a bimodule via (b.m.a)(m) = m (amb) for any m M, m M, a, b. If furthermore N is an C bimodule, Hom 1 (M,N) is considered as C bimodule through (b.ϕ.c)(m) = ϕ(mb)c, where b, ϕ Hom 1 (M,N), c C and m M. For a finite group G we consider any RG RG bimodule M as R(G G) module with (x,y) G G acting on m M as xmy 1 (and vice versa) y a complex we always mean a chain complex (that is, its differential has degree 1), and we implicitely consider any cochain complex X over any ring as chain complex through X n = X n, where n is an integer; note that then H n (X) = H n (X) for any integer n Let,, C be R algebras, X a complex of bimodules with differential δ, Y a complex of C bimodules with differential ǫ, and Z a complex of C bimodules with differential γ. We define X Y to be the complex of C bimodules whose component in degree n is equal to X i Y n i i Z and differential induced by the maps ( ) ( 1) i Id Xi ǫ n i : X δ i Id i Y n i X i Y n 1 i X i 1 Y n i. Yn i Similarly, we define Hom 1 (X,Z) to be the complex of C bimodules whose component in degree n is equal to with differential induced by the maps Π Hom 1(X i,z n i ) i Z (( 1) n Hom 1 (δ i+1,z n i ),Hom 1 (X i,γ n i )) For any integer i and any complex X with differential δ we denote by X[i] the complex whose component in degree n is equal to X n i, with differential given by

5 4 MRKUS LINCKELMNN the maps ( 1) i δ n i : X n i X n i 1. This sign convention has the property, that if X is a complex of left modules and i an integer, then the natural isomorphisms X n = Xn for any integer n induce an isomorphism of complexes [i] X = X[i], where [i] is the complex whose component in degree i is and zero in all other degrees, while if X is a complex of right modules we have to multiply the canonical isomorphisms X n = X n by the sign ( 1) ni in order to obtain an isomorphism of complexes X between the functors [i] [i] = X[i]. The above isomorphisms define natural isomorphisms category of complexes of bimodules., the degree i shift functor [i] and [i] on the The cone of a map of complexes f : X Y is the complex C(f) whose component in degree n is equal to X n 1 Y n, with differential induced by the maps X n Y n+1 X n 1 Y n whose restriction to X n is the map ( δ n,f n ) and whose restriction to Y n+1 is the map (0,ǫ n+1 ), where δ and ǫ are the differentials of X and Y, respectively. The cone comes along with obvious natural maps Y C(f) and C(f) X[1] For an R algebra we denote by Mod() the category of finitely generated modules, by C() the category of complexes of finitely generated modules and by K() its homotopy category. We denote by C b () and K b () the full subcategories of C() and K(), respectively, consisting of bounded complexes of modules. Recall that an module U is relatively R projective if it is isomorphic to a direct summand of an module of the form R V for some R module V; we denote by Mod() the R stable category of Mod(); that is, the objects of Mod() are the same as the objects of Mod(), and the morphisms are equivalence classes of homomorphisms for the relation declaring two homomorphisms between two modules to be equivalent if their difference factors through a relatively R projective module (thus, if R is a field, this is just the usual stable category of Mod()) We say that a finite-dimensional algebra over a field k is split if End (S) = k for every simple module, or, equivalently, if /J() is a direct sum of matrix algebras over k. 2 Transfer in Hochschild cohomology of symmetric algebras transfer associated to a suitable complex of bimodules has been considered by. Keller [12] for cyclic homology and S. ouc [4] for Hochschild homology. In the case of induction from a subalgebra, M. Feshbach has developed in [10] a transfer for the cohomology of Hopf algebras. We define here a transfer for Hochschild cohomology of symmetric algebras, using adjunction maps in a way which is similar to M. roué s definition in [5] of relative traces for symmetric algebras. The properties given in 2.11, 2.12 are analogous to results in [12] and [4] on cyclic and Hochschild homology, respectively. Let R be a commutative ring. Recall that an R algebra is symmetric if it is projective as R module and isomorphic to its R dual = Hom R (,R) as

6 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 5 bimodule. For any projective module M any choice of such a bimodule isomorphism =, or equivalently, of a symmetrizing form s on (cf. 6.3), gives rise to a natural isomorphism of functors (cf. 6.5) 2.1. Hom (M, ) = M from Mod() to Mod(R). Note that then M is projective as right module, since = is so. y naturality, this extends to complexes of bimodules as follows: if is another R algebra and X a bounded complex of bimodules which are projective as left modules, there is a natural isomorphism of functors 2.2. Hom 1 (X, ) = X from the category C() of complexes of modules to the category C() of complexes of modules mapping the subcategory C b () of bounded complexes of modules to the corresponding subcategory C b (). In particular, the functor X section 6 below), and we denote by 2.3. is a right adjoint to X ǫ X : X X, η X : X X (see [5, 1.12, 2.3] or the chain maps of complexes of bimodules and bimodules, respectively, representing the unit and counit of this adjunction. The adjunction maps in 2.3 depend on the choice of the symmetrizing form s on in the following way: 2.4. if s is another symmetrizing form on, there is a unique invertible element u Z() such that s = u.s; that is, s (a) = s(ua) for any a, and then the corresponding adjunction maps ǫ X, η X satisfy ǫ X = (Id X u.id X ) ǫ X, η X = u 1.η X. If in addition is symmetric and if all components of X are projective as right modules, too, the above discussion applies to X, and thus the functors X and X are both left and right adjoint to each other. In particular, choosing a symmetrizing form on we obtain chain maps of complexes of bimodules 2.5. ǫ X : X X, η X : X X

7 6 MRKUS LINCKELMNN representing the unit and counit of X being a right adjoint to X. It is possible to give explicit descriptions of ǫ M, η M (see [5] or 6.6). Note that the above discussion includes the case where X is an bimodule, considered then as complex concentrated in degree zero. Example 2.6 Let G be a finite group. The map sending λ (RG) to λ(x 1 )x RG is an isomorphism of RG RG bimodules; in particular, RG x G is symmetric and the symmetrizing form s (RG) corresponding to this bimodule isomorphism is the R linear map sending 1 G to 1 R and any non trivial element of G to 0 R. Let H be a subgroup of G and M = (RG) H the RG RH bimodule obtained from restricting the regular RG RG bimodule RG to RH on the right. That is, the functor M is the usual induction functor Ind G H. The R dual M of M RH is, via the above bimodule isomorphism (RG) = RG, naturally isomorphic to the RH RG bimodule H (RG)obtainedfromrestrictingtheregularRG RG bimodule RG to RH on the left, and the functor M is then naturally isomorphic to the RG restriction functor Res G H. In particular, M M = RG as RH RH bimodules, and, with this identification, the bimodule homomorphisms given by adjunction are as follows: ǫ M : RH RG is the inclusion, η M : RG RG RG is given by multiplication in RG, RH ǫ M : RG RG RG maps a RG to ax x 1, and RH x [G/H] η M : RG RH is the natural projection mapping x H to x and x G H to 0. Recall that if X is a bounded complex of bimodules, the degree zero component of X X is equal to X n X n n, where n runs over the set of integers. With some abusive identifications (see 6.9), the adjunction maps have the following properties (the formal proof is left to the reader): Proposition 2.7. Let,, C be symmetric R algebras, X, X bounded complexes of bimodules, Y a bounded complex of C bimodules, and suppose that all components of X, X, Y are projective as left and right modules. RG (i) We have ǫ X X = ǫ X +ǫ X and η X X = η X +η X. (ii) We have ǫ X Y = (Id Y ǫ X Id Y ) ǫ Y and η X = η X (Id X η Y Id X ). Y (iii) We have ǫ X = ( 1) n ǫ Xn and η X = ( 1) n η Xn, where n runs over the set n n of integers. The Hochschild cohomology of an R algebra whixh is projective as R module this is the ring HH () = Ext 0().

8 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 7 More explicitely, we denote by P a projective resolution of ; that is, P is a right bounded complex of projective bimodules endowed with a quasi-isomorphism µ : P, whereisviewedascomplexconcentratedindegreezero. Forexplicit calculations it may sometimes be helpful to use the standard projective resolution of as bimodule, which in degree n 0 is equal to (n+2) (the (n+2) fold tensorproduct of by itself over R) with differential δn : (n+2) (n+1) mapping a 0 a 1... a n+1 to ( 1) k a 0.. a k a k+1.. a n+1 (where 0 k n n > 0 and a j for 0 j n + 1), together with the map from the degree zero component to given by multiplication in. R y definition, HH n () is the cohomology in degree n of the cochain complex Hom 0(P,). y standard results of homological algebra, we have natural isomorphisms 2.8. HH n () = H n (Hom 0(P,)) = H n (Hom 0(P,P )) = Hom K( 0 )(P,P [n]). It is this latter form of the Hochschild cohomology that we are going to use most of the time, since it is convenient for dealing with the multiplicative structure in HH (), which is simply induced by composing chain maps. Observe that HH 0 () = Z() lies in the center of HH () and that HH n () is a module over Z() which is annihilated by the projective ideal Z pr () of Z() for n > 0 (cf. [5] ). We define the stable Hochschild cohomology HH () to be the quotient of HH () by the ideal generated by Z pr (); thus HH n () = HH n () for n > 0 and HH 0 () = HH 0 ()/Z pr ()HH 0 () = Z(). Recall another standard fact on homological algebra: if, are symmetric R algebras and X is a bounded complex of bimodules which are projective as left and right modules, the (total) complex X P X is a projective resolution of the complex X X; that is, it is right bounded (as P is so) consisting of projective bimodules and quasi-isomorphic to X X through the chain map Id X µ Id X, where µ : P is the given quasi-isomorphism as above. Therefore the adjunction map ǫ X : X X lifts to a chain map P X P X that we are going to denote by ǫ X again; this chain map is in general not unique, but unique up to homotopy. Similarly, η X lifts to a chain map X P X P still denoted by η X, which is again unique up to homotopy. With this notation we define now a transfer as follows:

9 8 MRKUS LINCKELMNN Definition 2.9Let, besymmetricr algebras,fixsymmetrizingformss, t, and let X be a bounded complex of bimodules which are projective as left and right modules. The transfer associated with X is the unique linear graded map t X : HH () HH () sending, for any n 0, the homotopy class [ζ] of a chain map ζ : P P [n] to the homotopy class [η X [n] (Id X ζ Id X ) ǫ X ] of the composition of chain maps ǫ P X X P X Id X ζ Id X X P [n] X η X[n] P [n]. The degree zero component t 0 X : HH0 () HH 0 () of t X induces a linear map Z() Z() through the natural isomorphisms HH 0 () = Z() and HH 0 () = Z() that we are going to denote abusively again by t 0 X, if no confusion arises. If M is an bimodule which is projective as left and right module, we denote by t M the transfer associated with the complex equal to M in degree zero and zero in all other degrees. Remark 2.10 The above definition depends on the choice of the symmetrizing forms s on and t on in the following way: if s and t are some other symmetrizing forms on and, there are unique invertible elements u Z() and v Z() such that s = u.s and t = v.t. It follows from 2.4 that the corresponding transfer map t X associated with the choice of the symmetrizing forms s and t instead of s and t satisfies t X([τ]) = u 1 t X (v[τ]) for any [τ] HH (). In the case of group algebras we assume always that the bimodule isomorphisms are the standard ones as in 2.6. The following proposition is an immediate consequence of 2.7: Proposition Let,, C be symmetric R algebras, X, X bounded complexes of bimodules and Y a bounded complex of C bimodules. ssume that all components of X, X, Y are projective as left and right modules. For any choice of symmetrizing forms on,, C we have: (i) t X X = t X +t X, (ii) t X Y = t X t Y, (iii) t X = n Z( 1) n t Xn. (iv) t X[i] = ( 1) i t X for any integer i. We list some further easy properties of the transfer, including in particular the statement analogous to [12, 2.4] saying that the transfer map t X depends only on the image of X in the Grothendieck group of the triangulated subcategory of K b ( 0 ) of bounded complexes whose components are projective as left and right modules:

10 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 9 Proposition Let, be symmetric R algebras with symmetrizing forms s, t, X, Y bounded complexes of bimodules which are projective as left and right modules, and let f : X Y be a chain map. Denote by C(f) the mapping cone of f. (i) The components of C(f) are projective as left and right modules, and we have t C(f) = t Y t X. (ii) If X is acyclic we have t X = 0. (iii) If f is a quasi-isomorphism we have t X = t Y. (iv) If 0 M M M 0 is an exact sequence of bimodules which are projective as left and right modules, we have t M = t M +t M. (v) If M is a projective bimodule then t M (HH n ()) = 0 for all n > 0 and t M (HH 0 ()) Z pr ()HH 0 (). Proof. (i) The component in degree i of C(f) is equal to X i 1 Y i and hence (i) follows from 2.11 (iii). (ii) If X is acyclic, the complex X P X is acyclic, right bounded and its components are projective (as bimodules), hence this complex splits and is therefore homotopic to zero, which implies the statement. (iii) If f is a quasi-isomorphism the cone C(f) is acyclic, and thus (iii) follows from (i) and (ii). (iv) is a particular case of (ii). Statement (v) follows from the fact that M P M is a projective resolution of the projective bimodule M M, hence a split complex which is homotopy equivalent to M M viewed as complex concentrated in degree zero. Remark 2.13 y [20, 2.5] the Hochschild cohomology ring is invariant under derived equivalences. For symmetric algebras this can be seen as follows: with the notation of 2.12, if and are derived equivalent, then J. Rickard proved in [20] that there is a bounded complex X of bimodules whose components are projective as left and right modules such that we have homotopy equivalences X X and X X as complexes of bimodules. Then t X and t X are mutually inverse ring isomorphisms between HH () and HH (), since all occurring adjunction maps are homotopy equivalences. Similarly, if M is an bimodule inducing a stable equivalence of Morita type between and (a concept due to roué [5]; see [13] for some properties of this notion), then t M and t M induce mutually inverse ring isomorphisms HH () = HH () by 2.12 (v). 3 Stable elements We provide here an abstract setting for the notion of stable elements in Hochschild cohomology of symmetric algebras and study its connections with transfer maps; this is intended to be a tool for dealing with the multiplicative structure

11 10 MRKUS LINCKELMNN of Hochschild cohomology rings. In section 5 we establish a link to the classical definition of stable elements in group cohomology as given in [9, Ch. XII, section 10]. Let R be a commutative ring. We keep the notation of the previous section for adjunction maps. Definition 3.1 Let, be symmetric R algebras with symmetrizing forms s, t, and X a bounded complex of bimodules which are projective as left and right modules. (i) We denote by π X the image η X ǫ X (1 ) in Z() of 1 under the composition of bimodule homomorphisms X X and call π X the relatively ǫx ηx X projective element of Z() (with respect to the choice of the symmetrizing forms s and t). (ii) If π X is invertible in Z() we denote by T X : HH () HH () the graded linear map defined by T X ([τ]) = π 1 X t X([τ]) for any [τ] HH (), and call T X the normalized transfer associated with X. (iii) n element [ζ] HH () is called X stable if there is [τ] HH () such that for any non negative integer n the following diagram is homotopy commutative P X ζ n Id X P [n] X X P Id X τ n X P [n] where ζ n and τ n are the components in degree n of ζ and τ, respectively, and where the horizontal arrows are given by the natural homotopy equivalences P X X P lifting the natural isomorphism X = X. We denote by HHX () the set of X stable elements in HH (). (iv) We say that an element z Z() is X stable, if the image of z in HH 0 () under the natural isomorphism Z() = HH 0 () lies in HHX 0 (), and denote by Z X () the set of X stable elements in Z(). Remark 3.2 The element π X defined in 3.1(i) depends on the choice of the symmetrizing forms s and t. In fact, denoting again abusively by t 0 X : Z() Z() the linear map induced by the degree zero component of t X via the natural isomorphisms Z() = HH 0 () and Z() = HH 0 (), we have π X = t 0 X(1 ). Thus if π X is the relatively X projective element with respect to another choice of symmetrizing forms s and t, there are unique invertible elements u Z(), v Z() such that s = u.s, t = v.t (cf. 2.4), and then by 2.10, we have

12 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS π X = u 1 t 0 X(v). Using 2.6 it is possible to construct examples where π X = 1 and π X = 0 (see 3.9 below), so not even the property of π X being invertible is independent of the choice of the symmetrizing forms. The previous formula shows in particular if π X is invertible, then setting s = (π X ) 1.s and t = t we have π X = 1. Observe that at the levels of complexes, the composition of chain maps P ǫx X P X P ηx is homotopic to either endomorphism of P induced by left or right multiplication with the element π X on P, since η X ǫ X lifts the endomorphism of given by left or right multiplication with the central element π X (cf ). The property of an element [ζ] HH () to be X stable does not depend on the choice of the symmetrizing forms on and. Note finally that an element z Z() is X stable if and only if there is an element y Z() such that the two chain endomorphisms of X induced by left multiplication with z and by right multiplication with y on X are homotopic. Lemma 3.3. Let, be symmetric R algebras with symmetrizing forms s, t, X a bounded complex of bimodules which are projective as left and right modules, and let [ζ] HH n () and [τ] HH n (), where n is a nonnegative integer. The diagram P X ζ Id X P [n] X X P Id X τ X P [n] is homotopy commutative if and only if any of the following diagrams is homotopy commutative: P ζ ǫ X X P Id X τ Id X X P [n] ǫ X [n] X P [n] X X P X Id X ζ Id X η X P τ X P [n] X η X [n] P [n]

13 12 MRKUS LINCKELMNN P τ Id X X X P Id X ζ P [n] X X P [n] P τ ǫ X X P X Id X ζ Id X P [n] ǫ X [n] X P [n] X X Id X τ Id X P X η X P ζ X P [n] X η X [n] P [n] Proof. If is homotopy commutative, tensoring this diagram by X and composing it with the commutative diagram P ζ Id P Id P ǫ X P X ζ Id X Id X X P [n] Id P [n] ǫ X P [n] X X shows that is homotopy commutative. y applying this argument in the appropriate way it follows that the homotopy commutativity of the diagrams 3.3.1, and is equivalent. Since the horizontal arrows in are homotopy equivalences we may reverse them, and then, again applying appropriate variations of the above argument shows that is homotopy commutative if and only if any of the diagrams 3.3.4, 3.3.5, is so. Lemma 3.4. Let, be symmetric R algebras with symmetrizing forms s, t and X a bounded complex of bimodules which are projective as left and right modules. Let [ζ] HH X () and [τ] HH () such that the diagram is homotopy commutative for all nonnegative integers n. (i) We have [τ] HH X (). (ii) We have t X ([τ]) = π X [ζ]. (iii) For any [σ] HH () we have t X ([τ][σ]) = [ζ]t X ([σ]) and t X ([σ][τ]) = t X ([σ])[ζ]. (iv) We have t X X([τ]) = π X X[τ].

14 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 13 Proof. Statement (i) follows from the fact that is homotopy commutative if and only if is so. Using we get t X ([τ]) = [η X ǫ X ζ], and since η X ǫ X is homotopic to multiplication by π X on P we obtain (ii). Similarly, composing the diagrams and with the diagram defining t X ([τ]) in each component yields the two equalities in (iii). For (iv) apply t X to the equality (ii) and use then the analogue of (iii) with X instead of X, together with the fact that π X t 0 X X (1 ) = t 0 X (π X) by and 2.11(ii). X = Proposition 3.5. Let, be symmetric R algebras with symmetrizing forms s, t, and X a bounded complex of bimodules whose components are projective as left and right modules. (i) The set HH X () of X stable elements in HH () is a graded subalgebra of HH (); in particular, Z X () is a subalgebra of Z(). (ii) The space Im(t X ) is a graded HH X () HH X () subbimodule of HH (). (iii) We have t X (HH X ()) = π XHH X (). (iv) For every direct summand X of the complex X we have HH X () HH X (). Proof. Clearly HHX () is a graded R submodule of HH (), and the composition of diagrams of the form shows that indeed HHX () is a ring, which shows (i). Statement (ii) follows from 3.4(iii), and (iii) is a consequence of 3.4(i) and 3.4(ii). Moreover, if is homotopy commutative, decomposing the complex X into a direct sum of complexes yields a decomposition of the diagram into a direct sum of homotopy commutative diagrams, which implies (iv). Theorem 3.6. Let, be symmetric R algebras with symmetrizing forms s, t, and X be a bounded complex of bimodules which are projective as left and right modules. (i) If π X is invertible, the normalized transfer T X induces a graded surjective R algebra homomorphism R X : HH X () HH X(). (ii) If π X is invertible and s, t are symmetrizing forms such that the corresponding relatively X projective element π X is again invertible, then, denoting by R X the algebra homomorphism induced by the normalized transfer T X associated with the choice of s, t instead of s, t, we have R X = R X. (iii) If both π X and π X are invertible then R X and R X are mutually inverse R algebra isomorphisms HHX () = HHX(). Proof. (i) y 3.4 (ii) and 3.4 (iii), the map R X is indeed an R algebra homomorphism, and by 3.5 (iii) it is surjective. (ii) Let u Z() and v Z() be the unique elements such that s = u.s and t = v.t. pplying 3.4 to [τ] HHX () yields t0 X (v)t X([τ]) = π X t X (v[τ]). Multiplying this equation by u 1 shows then that π X t X([τ]) = π X t X ([τ]) by Since both π X and π X are invertible it follows that R X([τ]) = R X ([τ]). Statement (iii) is an easy verification, using again 3.4(i) and 3.4(ii).

15 14 MRKUS LINCKELMNN Proposition 3.7. Let,, C be symmetric R algebras, X a bounded complex of bimodules, and Y a bounded complex of C bimodules. Suppose that all components of both X and Y are projective as left and right modules. Let n be a nonnegative integer, [ζ] HH n () and [γ] HH n (C) making the following diagram homotopy commutative: P ζ ǫ Y X X Y C P C C Id X Y X γ Id Y X Y P [n] X Y P C [n] Y X ǫ Y X [n] C C Suppose there is z Z() such that left multiplication by z and right multiplication by π Y on X induce homotopic endomorphisms of X. Then the following diagram is homotopy commutative: z.ζ P ǫ X X P [n] ǫ X [n] P Id X t Y (γ) Id X X X P [n] X In particular, z[ζ] is X stable and t Y ([γ]) is X stable. Proof. y 2.7(ii) we have a homotopy commutative diagram P Id P ǫ X X P Id X ǫ Y Id X X P ǫ Y X X Y P C Y X C C (with the usual identification (X Y) = Y X, see 6.7). Moreover, by 2.7(ii) X = (Id X η Y ǫ Y Id X ) ǫ X, and as η Y ǫ Y is we have (Id X η Y Id X ) ǫ Y homotopic to right (or left) multiplication by π Y on P, it follows from the hypothesis on z, that we have a homotopy commutative diagram z.id P P ǫ Y X X P ǫ X Y C P C Y C Id X η Y Id X X P X Now shift the diagram by the degree n and match it (vertically) together with the diagrams and 3.7.3; this yields the diagram as required. X

16 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 15 Corollary 3.8. Let,, C, X and Y be as in 3.7. If π Y is invertible in Z() we have HHX Y () HH X () and T Y(HHY X (C)) HH X (). Proof. Recall that the property of being stable with respect to some complex is independent of the choice of symmetrizing forms; thus, as π Y is invertible we may in fact assume that π Y = 1 by and 3.6(ii). Then z = 1 fulfills the hypothesis in 3.7, and thus 3.8 follows from 3.7. Example 3.9 Let G be a finite group, H a subgroup of G and M = (RG) H. With respect to the canonical symmetrizing forms on RG and RH we have π M = [G : H].1 RG and π M = 1 RH as follows from the explicit description of the adjunction maps in 2.6. If there is z Z(G) such that z lies not in H, then s = z.s is again a symmetrizing form on RG, and now the relatively projective elements with respect to s and the canonical form on RH are π M = [G : H].z and π M = 0 RH by Transfer and group cohomology Let R be a commutative ring. The cohomology of a finite group G with coefficients in R is the ring (cf. [2]) 4.1. H (G,R) = Ext RG(R,R), where R is considered as RG module with the trivial action of the elements of G on R. gain, it is well-known that there is a natural isomorphism 4.2. H n (G,R) = Hom K(RG) (P R,P R [n]) for any nonnegative integer n, where P R is a projective resolution of the trivial RG module R. If we denote by G the diagonal subgroup G = {(x,x)} x G of G G, there is a unique isomorphism of RG RG bimodules 4.3. Ind G G G (R) = RG mapping (x,y) 1 R to xy 1, where x, y G (with our convention of considering the R(G G) module Ind G G G (R) as RG RG bimodule). Thus the complex Ind G G G (P R) is a projective resolution of RG; in particular, the isomorphism in 4.3 lifts to a homotopy equivalence of complexes of RG RG bimodules 4.4. Ind G G G (P R) P RG, which is unique up to homotopy, where here P RG is a projective resolution of RG as RG RG bimodule as above. The following is well-known (and we leave the proof to the reader):

17 16 MRKUS LINCKELMNN Proposition 4.5. Let G be a finite group and P R a projective resolution of the trivial RG module R. The map sending ζ Hom C(RG) (P R,P R [n]) to Ind G G G (ζ), where n is a nonnegative integer, induces an injective R algebra homomorphism δ G : H (G,R) HH (RG). Recall that for any subgroup H of G the usual transfer map (see e.g. [2]) t G H : H (H,R) H (G,R) can be defined as follows: since the restriction to H of a projective resolution P R of the trivial RG module is a projective resolution of the trivial RH module, any element of H n (H,R) can be represented by a chain map τ : Res G H (P R) Res G H (P R[n]), from which we obtain a chain map TrH G(τ) : P R P R [n] defined by TrH G(τ)(a) = xτ(x 1 a), where a P R, and then x [G/H] t G H ([τ]) = [TrG H (τ)]. Moreover, if ϕ : H G is any injective group homomorphism, we denote by res ϕ : H (G,R) H (H,R) the graded linear map induced by restriction through ϕ. The usual transfer and restriction for group cohomology are without any surprise compatible with the corresponding transfers in Hochschild cohomology through the algebra homomorphisms in 4.5 (recall that we consider RG with the canonical symmetrizing form mapping 1 G to 1 R and any non trivial element of G to 0 R ): Proposition 4.6. Let G be a finite group and H a subgroup of G. The following diagram is commutative: H (H,R) δ H t G H H (G,R) δ G HH (RH) t (RG)H HH (RG) Proof. Let P R be a projective resolution of the trivial RG module R. Let n be a nonnegative integer and τ Hom C(RH) (P R,P R [n]). Then TrH G followed by IndG G G maps τ to the chain map and, the other way round, Ind H H H t (RG)H maps τ to the chain map Ind G G G (P R) Ind G G Ind G G G (TrG H (τ)) G (P R[n]) followed by the diagram defining the transfer Ind G G G (P R) Ind G G H (P R) Ind G G H (τ) Ind G G H (P R[n]) Ind G G G (P R[n]), where the first and last map are induced by the appropriate adjunction maps. Using the explicit description in 2.6 of these adjunction maps shows that the two chain maps defined in and coincide, which completes the proof.

18 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 17 Proposition 4.7. Let G, H be finite groups and ϕ : H G an injective group homomorphism. The following diagram is commutative: H (G,R) δ G res ϕ H (H,R) δ H HH (RG) t ϕ (RG) HH (RH) Proof. We may assume that H is a subgroup of G and that ϕ is the inclusion. Then 4.7 follows from the explicit description of the adjunction maps corresponding to the transfer associated with the bimodule H (RG) in 2.6. Proposition 4.8. Let G be a finite group and H a subgroup of G. (i) We have a natural isomorphism of functors Res G G G H IndG G G = Ind G H H Res G H from Mod(R G) to Mod(R(G H)). (ii) For any nonnegative integer n and any chain map ζ Hom C(RG) (P R,P R [n]) we have a commutative diagram of complexes of RG RH bimodules Ind G G G (P = R) (RG) H RG RG RH δ G (ζ) Id RG Ind H H H (P R) Id RG δ H (ζ) Ind G G G (P R[n]) (RG) H RG Ind H H RG = H (P R[n]) RH (iii) We have Im(δ G ) HH (RG) H (RG). Proof. Statement (i) is a particular case of the Mackey formula. Through our convention of considering RG RH bimodules as R(G H) modules the left column in the diagram of (ii) is obtained from applying the functor Res G G G H IndG G G to the chain map ζ, and the right column is obtained from applying the functor Ind G H H Res G H to ζ, thus (ii) follows from (i). The horizontal isomorphisms in the commutative diagram in (ii) yield precisely the homotopy equivalences of the diagram applied to (RG) H instead of X, which shows (iii). 5 The cohomology ring of a p block of a finite group In this section we fix a prime p and a complete discrete valuation ring O having a residue field k = O/J(O) of characteristic p. We assume that either O has characteristic zero or O = k. We adopt the following abuse of notation: if n, m Z such that n 0 and v p (m) v p (n), where v p is the p adic valuation, we set m n 1 O = (m 1 O )(n 1 O ) 1, where m, n Z such that (n,p) = 1 and m n = m Recall that a block of a finite group G is a primitive idempotent b in Z(OG), the algebra OGb is called block algebra of b, and a defect group of b is a minimal subgroup n.

19 18 MRKUS LINCKELMNN P of G such that the map OGb OGb OGb given by multiplication in OGb OP splits as homomorphism of OGb OGb bimodules. We refer to [1], [17], [18] for recalls on pointed groups, fusion in block algebras and rauer pairs. See [14, section 6] for a short review (and further references) on fusion which is pretty much adapted to our needs in this section. detailed account of the material on block theory we use here can be found in Thévenaz book [23]. s mentioned in the introduction, the cohomology ring H (G,O) of G has three algebraic characterizations: a global characterization as the ring Ext OG (O,O), a local characterizationassubringofstableelementsinthecohomologyringh (P,O) of a Sylow-p subgroup P of G, and a characterization as a subring of relatively G projective elements of the Hochschild cohomology ring HH (OG) by 4.5. The two latter characterizations have obvious generalizations to any p block b of a finite group G, replacing P by a defect group of b and the Hochschild cohomology of OG by that of OGb. These two notions still coincide as we are going to show in this section, while it is not clear at this stage, whether there is an analogous global description of the cohomology ring of the block b, as there need not be an augmentation (not even the source algebras of b need have an augmentation). Definition 5.1 Let G be a finite group, b a block of G, P γ a defect pointed group of G {b}. Let i γ and denote for any subgroup Q of P by e Q the unique block of kc G (Q) satisfying r Q (i)e Q 0. The cohomology ring of the block b of G associated with P γ is the subring H (G,b,P γ ) of H (P,O) which consists of all [ζ] H (P,O) satisfying res ϕ ([ζ]) = res P Q ([ζ]) for any subgroup Q of P and any automorphism ϕ of Q induced by conjugation with an element of N G (Q,e Q ). Since all defect pointed groups of G {b} are conjugate, the above ring is, up to isomorphism, independent of the choice of P γ, and we will write H (G,b) instead of H (G,b,P γ ) if the choice of P γ is obvious from the context. Remarks It follows from lperin s fusion lemma for rauer pairs (cf. [23, (48.3)]) that if an element [ζ] H (P,O) belongs to H (G,b,P γ ) then in fact res ϕ ([ζ]) = res P Q ([ζ]) for any injective group homomorphism ϕ : Q P induced by conjugation with an element x G satisfying x (Q,e Q ) (P,e P ), or equivalently, ϕ E G ((Q,e Q ),(P,e P )) (cf. [18]). lperin s fusion lemma implies also that in order to check whether an element [ζ] of H (P,O) belongs to H (G,b,P γ ) it suffices to check that res ϕ ([ζ]) = res P Q ([ζ]) for any Q P such that (Q,e Q) is self-centralizing (that is, e Q has Z(Q) as defect group) and for any automorphism ϕ of Q induced by conjugation with an element of N G (Q,e Q ) If b 0 is the principal block of G then for any subgroup Q of P the block e Q is the principal block of kc G (Q), hence N G (Q,e Q ) = N G (Q), and therefore, by [9, Ch. XII, 10.1], we have H (G,b,P γ ) = H (G,O),

20 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 19 the usual cohomology ring of G with coefficients in O If the defect group P of b is abelian, then the inertial quotient E = N G (P γ )/C G (P) controls fusion (cf. [1] or [23, (49.6)]) and hence we have H (G,b,P γ ) = H (P,O) E = H (P E,O) The cohomology ring H (G,b,P γ ) is in fact an invariant of the source algebra iogi of b, where i γ. This is because if (Q,e Q ) is self-centralizing, there is a unique local point δ of Q on iogi (cf. [23, (41.1)]) and by a theorem of Puig [18, 3.1], the group E G (Q,e Q ) = N G (Q,e Q )/QC G (Q) viewed as subgroup of the outer automorphism group of Q coincides with the group of iogi fusion F iogi (Q δ ) (see [18] or the appendix of [14] for a brief account on these ideas). One might therefore as well write H (iogi) = H (G,b,P γ ) and call this the cohomology ring of iogi. More generally, we can define the cohomology ring of an interior P algebra over O to be the subring of elements [ζ] in H (P,O) satisfying res P Q ([ζ]) = res ϕ([ζ]) whenever Q is a subgroup of P and ϕ : Q P an injective group homomorphism such that the class of ϕ modulo inner automorphisms of P belongs to the set of fusion F (Q,P) = F (Q δ,p γ ), where γ and δ run over the sets of local points of P and γ,δ Q on, respectively. The next lemma provides the technicalities in order to establish a connection between stable elements in group cohomology in the sense of [9, Ch. XII, section 10] and the stability notion developed in section 3. Lemma 5.3. Let P be a finite group, Q a subgroup of P and ϕ : Q P an injective group homomorphism. Let n be a nonnegative integer and [ζ] H n (P,O). The following statements are equivalent. (i) We have res ϕ ([ζ]) = res P Q ([ζ]). (ii) The diagram δ Q (ζ) P OQ ǫ ϕ ϕ (P OP ) ϕ δ P (ζ) P OQ [n] ǫ ϕ [n] ϕ(p OP [n]) ϕ is homotopy commutative, where ǫ ϕ is a chain map lifting the homomorphism of OQ OQ bimodules OQ ϕ (OP) ϕ sending u Q to ϕ(u). (iii) The diagram δ P (ζ) P OP ǫ OP OQ ϕ (OP) (OP) ϕ P OP OQ OQ ϕ (OP) Id (OP)ϕ δ P(ζ) Id ϕ (OP) P OP [n] ǫ OP OQ ϕ (OP)[n] (OP) ϕ P OP [n] OQ OQ ϕ (OP) is homotopy commutative.

21 20 MRKUS LINCKELMNN Moreover, if the above statements hold then t OP OQ ϕ (OP)(δ P ([ζ])) = [P : Q]δ P ([ζ]); in particular, π OP OQ ϕ (OP) = [P : Q]1 OP. Proof. Denote by P O a projective resolution of the trivial OP module O. The equalityres ϕ ([ζ]) = res P Q ([ζ])holdsifandonlyifthediagramofcomplexesofoq modules res P Q (P O) res P Q (ζ) res P Q (P O[n]) res ϕ (P O ) res ϕ (ζ) res ϕ (P O [n]) is homotopy commutative, where the horizontal maps lift the identity on O viewed as homomorphism of OQ modules (this makes sense, since the complexes res P Q (P O) and res ϕ (P O ) are both projective resolutions of the trivial OQ module). pply now the induction functor Ind Q Q Q to this diagram (where we consider an OQ moduleaso Q modulethroughtheobviousisomorphism Q = Q); asforany OP module U there is a natural isomorphism Ind Q Q Q ( ϕu) = ϕ (Ind ϕ(q) ϕ(q) ϕ(q) (U)) ϕ of OQ OQ bimodules mapping (y,y ) u to (ϕ(y),ϕ(y )) u, where x,x Q and u U, we obtain a diagram Q (P O) ϕ (Ind ϕ(q) ϕ(q) ϕ(q) (P O )) ϕ δ ϕ(q) (ζ) Ind Q Q δ Q (ζ) Q (P O[n]) ϕ (Ind ϕ(q) ϕ(q) (P O [n])) ϕ Ind Q Q ϕ(q) (where we write P O instead of res P Q (P O) and res P ϕ(q) (P O)). Since the identity functor on the category of O Q modules is isomorphic to a direct summand of the functor Res Q Q Q IndQ Q Q it follows that the diagram is homotopy commutative if and only is so. Now the functor taking an OP module U to the OQ OQ bimodule (U)) ϕ is isomorphic to a direct summand of the functor sending ϕ(ind ϕ(q) ϕ(q) ϕ(q) U to ϕ (Ind P P P (U)) ϕ via the natural transformation sending (ϕ(y),ϕ(y )) u to (ϕ(y),ϕ(y )) u, where y,y Q and u U (this is in fact a particular case of Mackey s formula). Composing with this natural transformation yields therefore a diagram Ind Q Q Q (P O) ϕ (Ind P P P (P O)) ϕ δ P (ζ) δ Q (ζ) Ind Q Q Q (P O[n]) ϕ (Ind P P P (P O[n])) ϕ which is homotopy commutative if and only if is so, since the horizontal arrows identify the left column to a direct summand of the right column, up to homotopy. Moreover, through the identification ϕ (OP) Ind P P P (P O) (OP) ϕ OP OP = ϕ (Ind P P P (P O)) ϕ the horizontal maps are precisely the adjunction map ǫ (OP)ϕ

22 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 21 and its shift by the degree n, since they lift the bimodule homomorphism OQ = Ind Q Q Q (O) ϕ(ind P P P (O)) ϕ = ϕ (OP) ϕ sending u Q to ϕ(u). This shows the equivalence of (i) and (ii). Observe in particular that this shows also that the diagram in (ii) is homotopy commutative if we take for ϕ the inclusion homomorphism Q P. y and 3.3.5, the diagram in statement (ii) is homotopy commutative if and only if the diagram δ P (ζ) P OP (OP) ϕ P OQ OQ OQ ϕ (OP) Id δ Q (ζ) Id P OP [n] (OP) ϕ P OQ [n] OQ OQ ϕ (OP) is homotopy commutative. Consider now the diagram in (ii) with the inclusion Q P instead of the homomorphism ϕ, tensor this diagram with (OP) ϕ OQ OQ ϕ (OP) and compose it with the diagram above; this yields the diagram of statement (iii) and hence the equivalence of (ii) and (iii). The last statement is a consequence of 2.11(ii), 4.6 and 4.7 together with the wellknown fact, that on ordinary group cohomology, restriction to a subgroup followed by the transfer is just multiplication by the index of the subgroup. Proposition 5.4. Let G be a finite group, b a block of G, P γ a defect pointed group of G {b} and let i γ. Consider iogi as OP OP bimodule and let [ζ] H (G,b,P γ ). (i) We have t iogi (δ P ([ζ])) = rk O(iOGi) P δ P ([ζ]); in particular, π iogi = rk O (iogi) P 1 OP. (ii) For any nonnegative integer n, the following diagram is homotopy commutative: δ P (ζ n ) P OP ǫ iogi iogi P OP iogi OP OP Id δ P (ζ n ) Id P OP [n] ǫ iogi [n] iogi P OP [n] iogi OP OP where ζ n is the degree n component of ζ; in particular, δ P ([ζ]) is iogi stable. Proof. y [14, 6.6], any indecomposable direct summand of iogi is isomorphic to OP OQ ϕ (OP) for some subgroup Q of P and some injective group homomorphism ϕ : Q P such that ϕ E G ((Q,e Q ),(P,e P )) where the notation is as in 5.1. Thus we have res ϕ ([ζ]) = res P Q ([ζ]). Since rk O(OP OQ ϕ (OP)) = [P : Q] P, statement (i) follows from the last statement in 5.3 and 2.11(ii), and statement (ii) is a consequence of the equivalence of 5.3(i) and 5.3(ii) together with the additivity 2.7(i) of adjunction maps.

23 22 MRKUS LINCKELMNN It is possible to make the adjunction maps associated with OGi and its dual iog explicit (we need this for the calculus of the relative projective elements): Lemma 5.5. Let G be a finite group, b a block of G, P γ a defect pointed group of G {b} and let i γ. The isomorphism (OG) = OG mapping the canonical symmetrizing form s to 1 OG induces an isomorphism of OP OGb bimodules (OGi) = iog and multiplication in OGb induces an isomorphism iog OGi = iogi. With the identifications through these isomorphisms, the adjunction maps associated with OGi and iog are given as follows: OGb ǫ OGi : OP iogi maps u P to ui; η OGi : OGi iog OGb is induced by multiplication in OGb; OP ǫ iog : OGb OGi iog maps a OGb to axi ix 1 ; OP x [G/P] η iog : iogi OP maps c iogi to s(cu 1 )u. Moreover, we have π OGi = Tr G P (i) and π iog = s(i)1 OP = rk O(iOG) G 1 OP. u P Proof. The fact that the first two maps are as stated follows, for instance, from the explicit description of adjunction maps in 6.6, and then the remaining two maps are obtained by the duality 6.8 (or again from the explicit descriptions in 6.6). It follows that π OGi = TrP G(i) and π iog = s(iu 1 )u. In order to compute the latter expression we may assume that O has characteristic zero; indeed this expression is invariant under extensions, so we may assume that k is perfect, and then it suffices to observe that k occurs as residue field of some complete discrete valuation ring (cf. [21]). Then G s is equal to the regular character of OG; thus G s(iu 1 ) is the value of the character of the projective OP module iog at the element u 1, thus equal to rk O (iog) if u = 1 and zero otherwise. The lemma follows. u P The next theorem is now the announced embedding of H (G,b,P γ ) into HH (OGb): Theorem 5.6. Let G be a finite group, b a block of G, P γ a defect pointed group of G {b} and let i γ. ssume that (iogi)(p) is a split k algebra. Consider OGi and iog as OGb OP bimodule and OP OGb bimodule, respectively. (i) We have π OGi = Tr G P (i) Z(OGb) and π iog = rk O(iOG) G 1 OP O 1 OP. (ii) If [ζ] H (G,b,P γ ) then δ P ([ζ]) is iog stable in HH (OP). (iii) The map T OGi δ P induces an injective O algebra homomorphism H (G,b,P γ ) HH OGi(OGb). Proof. (i) It follows from the previous lemma that the relative projective elements are as stated in (i). Since (iogi)(p) is split we may apply [16, Prop. 1] (see also [22, 9.3]), which shows that π OGi is invertible. In order to show that π iog is invertible, we may assume that O = k. Then ker(r P ) ker(s), where s is the canonical

24 TRNSFER IN HOCHSCHILD COHOMOLOGY OF LOCKS 23 symmetrizing form on kg, and therefore s(i) = s(r P (i)). Now r P (i) is a primitive idempotent in kc G (P), and applying the last statement of 5.5 to r P (i) instead of i showsthats(r P (i)) = dim k(r P (i)kc G (P)) C G (P). Thislastexpressionisnonzeroink, since r P (i)kc G (P) is, up to isomorphism, the unique projective indecomposable module of the nilpotent block e P of kc G (P) (cf. [7]), which is split by the assumptions. (ii)sinceiogi = iog OGiandπ OGi isinvertible, wecanapply3.8to = C = OGb OP, = OGb, X = iog and Y = OGi. Then 5.4(ii) implies that δ P (H (G,b,P γ )) HHiOGi (OP) HH iog (OP). (iii) follows from (ii) together with 3.6. The last theorem of this section is a generalization of the well-known fact that if a subgroup of a finite group controls p fusion, then the restriction to this subgroup induces an isomorphism on mod p cohomology. Theorem 5.7. Let G, H be finite groups, b, c blocks of G, H, respectively, having a common defect group P, let γ and δ be local points of P on OGb and OHc, respectively, and let i γ and j δ. For any subgroup Q of P, denote by e Q and f Q the unique blocks of kc G (Q) and kc H (Q), respectively, satisfying rq G(i)e Q 0 and rq H(j)f Q 0. Let X be a bounded complex of OGb OHc bimodules whose components are direct sums of direct summands of the bimodules OGi joh, with Q running over the set OQ of subgroups of P and set Y = ixj, considered as complex of OP OP bimodules. (i) We have π Y = π Y = ( 1) n rk O (Y n ) P 1 OP. n Z (ii) We have t Y (δ P (H (P,O))) δ P (H (P,O)). If moreover E G ((Q,e Q ),(P,e P )) E H ((Q,f Q ),(P,f P )) for any subgroup Q of P, then the following hold: (iii) We have H (H,c,P δ ) H (G,b,P γ ) and δ P (H (H,c,P δ )) HH Y (OP) HH Y (OP). (iv) We have t Y (δ P ([ζ])) = π Y δ P ([ζ]) for any [ζ] H (H,c,P δ ); in particular, if π Y is invertible then T Y restricts to the identity on H (H,c,P δ ) through δ P. Proof. y the hypotheses on the components of X, for any integer n, any indecomposable direct summand W of Y n or Yn is of the form OP OQ ϕ (OP) for some subgroup Q of P and some injective group homomorphism ϕ : Q P, thus (i) follows from the last statement in 5.3 and 2.11(iii). Moreover, by 4.6 and 4.7 we have t W (δ P ([ζ])) = δ P (t P Q res ϕ[ζ]) for any [ζ] H (P,O) which implies (ii). If E G ((Q,e Q ),(P,e P )) E H ((Q,f Q ),(P,f P )) for any subgroup Q of P, then H (H,c,P δ ) H (G,b,P γ ) by the definition 5.1. lso, with W and ϕ as before, we have now ϕ E H ((Q,f Q ),(P,f P )). This shows (iv), using again the last statement of 5.3. y matching together the diagrams that we get from 5.3(iii) we obtain (iii). Remark 5.8 With the notation and hypotheses of 5.7, if E G ((Q,e Q ),(P,e P )) = E H ((Q,f Q ),(P,f P )) for any subgroup Q in P then H (G,b,P γ ) = H (H,c,P δ ). One